Measurement-Aware Computational Modeling for Optical Spectrometry in Scattering-Dominated Systems

Abstract

Computational modeling is widely used to interpret optical spectrometric data; however, most prevailing approaches extrapolate isolated-particle optical cross sections to finite concentrations without explicitly accounting for photon transport, multiple scattering, and instrument geometry. In scattering-dominated systems, this simplification introduces systematic discrepancy between theoretical predictions and measured extinction and polarization signals, often misattributed to intrinsic material changes. Here, we develop a measurement-aware, physics-guided computational framework that integrates electrodynamic theory with Monte Carlo photon transport simulations and supervised least-squares parameter estimation under realistic experimental conditions. Rather than relying on direct extrapolation of single-particle responses, the framework forward-simulates photon trajectories within a finite sample volume, explicitly incorporating multiple scattering and detector collection constraints. A limited set of effective parameters, including collection geometry and refractive index, is inferred through cross-validated regression to enable quantitative comparison with experimental measurements. Using spherical polystyrene suspensions spanning three size regimes (100, 380, and 1060 nm) and broad concentration ranges, the model reproduces experimentally measured UV–Vis extinction spectra exhibiting pronounced nonlinear concentration dependence. The framework captures regimes in which Beer–Lambert extrapolation predicts optical densities of 5–9, whereas measured signals remain below 4 within the instrument’s linear dynamic range due to geometry-dependent photon redistribution. Spatial- and polarization-resolved simulations further show that multiple scattering reshapes both intensity and polarization in an orientation-, concentration-, and aperture-dependent manner, fundamentally altering the mapping between concentration and detected signal. By explicitly incorporating transport effects as part of the measurement process, this approach enables consistent extraction of effective refractive indices across independent particle sizes and data sets. The results establish a quantitative framework for reconciling theory with experiments in scattering-dominated optical systems and provide a scalable pathway for modeling more complex absorbing and emissive materials.

1. Introduction

Optical spectrometry is among the most widely used quantitative measurement techniques for determining the size, composition, and structure of materials across chemistry, biology, and nanoscience. − Under ideal conditions and within an instrument’s linear dynamic range, measured spectra are often interpreted as direct representations of intrinsic photophysical properties. Accordingly, for a given analyte, its composition, size, and shape are commonly inferred directly from the measured spectrum. − However, in nanoscale and heterogeneous systems, particularly those exhibiting significant scattering efficiency, measured signals are shaped not only by intrinsic material response but also by photon transport, multiple scattering, boundary interactions, and instrument geometry. ,− As a result, the relationship between experimentally detected intensity and intrinsic optical cross sections from isolated-particle predictions can deviate substantially, even when measurements exhibit high signal-to-noise ratios and operate within nominal linear regimes. ,−

A common practice in optical spectroscopy is to interpret ensemble measurements by comparing experimentally measured spectra with simulations derived from isolated, single-particle optical responses. ,, In UV–Vis spectrometry, this approach is typically extended by extrapolating extinction, absorption, or scattering cross sections to finite concentrations using Beer–Lambert-type assumptions. , While such strategies are appropriate for optically dilute systems dominated by absorption, they become unreliable in scattering-dominated or optically dense media, where photon trajectories are not terminal but repeatedly redistributed through multiple scattering events. ,,− ,, Under these conditions, deviations between measured spectra and isolated-particle predictions are frequently attributed to aggregation, size variation, or compositional change, even when the discrepancies arise from unmodeled photon transport and detection geometry. , Consequently, conventional extrapolation can introduce systematic inference errors in extracted optical parameters despite measurements appearing to remain within nominal instrumental linearity. ,,

These limitations arise because optical measurements in scattering media inherently involve cascading photon–matter interactions, in which incident photons undergo multiple scattering, boundary reflections, and redistribution before reaching the detector. ,,, Unlike purely absorbing systems, where photon–matter interactions are effectively terminal, scattering-dominated systems involve coupled transport processes that modify both the spatial distribution and polarization state of the detected light. , When absorption, scattering, and detection geometry interact within a finite sample volume, the measured signal becomes a convolution of intrinsic optical response and photon transport pathways rather than a direct scaling of single-particle cross sections. − Consequently, accurate interpretation of ensemble spectra requires explicit incorporation of photon propagation, boundary conditions, and detector collection constraints as integral components of the measurement model rather than as post hoc corrections. −

Because of these issues, researchers are continuously working to improve the precision and reliability of spectroscopic measurements and their interpretation. − Complementary to experimental efforts, a large body of theoretical modeling has been developed to compute optical spectra (absorption, scattering, extinction) of nanoparticles or small aggregates. − Classic electrodynamics (e.g., Mie theory and T-matrix methods , for spheres) and numerical methods (e.g., discrete-dipole approximation, , finite-difference time-domain method) can provide mechanistic insight and predict how spectral features shift with size, shape, environment or material dielectric function. In colloidal ensembles, methods combining Mie theory with size-distribution fitting or machine learning tools have been developed to extract particle size and concentration from UV–Vis spectra, work which underscores both the promise and the limitations of “inverse” spectral interpretation based purely on simple models. ,−

To address this measurement modeling gap, we propose a computational framework (Figure ) that integrates electrodynamic theory with Monte Carlo photon-transport simulations − and machine-learning based instrumental parameter optimization under realistic experimental boundary conditions. Specifically, the framework is based on a physics-guided forward model (Monte Carlo photon transport coupled with Mie theory calculations), while the optimization step is used to estimate a limited set of effective parameters that are not directly accessible or fully captured in the current formulation. In this sense, the approach is not a purely empirical curve-fitting procedure, but rather a physics-constrained parameter estimation framework. While Monte Carlo approaches have been widely used to model scattered light propagation, − their ability to quantitatively reproduce experimental measurements by incorporating realistic experimental parameters remains limited. Rather than extrapolating isolated-particle cross sections, the model forward-simulates photon trajectories through a finite sample volume, explicitly incorporating multiple scattering, cuvette geometry, and detector collection constraints. To our knowledge, this represents the first unified framework capable of simultaneously reproducing experimentally measured UV–Vis spectra across multiple particle sizes, concentrations, and wavelengths within a measurement-aware model. In this approach, ensemble extinction and polarization signals emerge directly from the coupled particle response and photon transport, enabling quantitative comparison with experimentally measured spectra.

1.

Flowchart of the iterative computational spectrometry framework. The workflow comprises three main sections: the theoretical model (blue region), experimental measurement (yellow region), and parameter optimization (gray region), with coupling between them. The process begins with initialization of input parameters, including excitation geometry, sample morphology and composition, and intrinsic optical constants. The blue region represents the forward physical model, where cascading photon–matter interactions in the cuvette are simulated to generate polarization- and wavelength-resolved photon distributions, yielding predicted spectra. These predicted spectra are compared with experimentally measured spectra. The gray region denotes an iterative parameter optimization loop, in which a machine-learning-assisted least-squares fitting algorithm updates model parameters within physically constrained bounds based on prediction error. This feedback process continues until the error falls below a predefined tolerance. Upon convergence, the simulation terminates and the final spectra are saved (green region). Red text highlights components associated with parameter optimization and iterative refinement.

As a foundational case, we focus on scattering-dominated particle suspensions, which provide a minimal yet experimentally stringent testbed for evaluating geometry-induced spectral distortion. By isolating systems with negligible intrinsic absorption or emission, we establish a baseline for assessing how photon transport alone governs deviations of the detected UV–Vis spectra from Beer–Lambert behavior before extending the approach to increasingly complex absorbing and emissive systems.

2. Experimental Section

2.1. Chemicals and Equipment

Carboxylate-functionalized polystyrene nanoparticles (Polybead PSPs) were obtained from Polysciences, Inc. The TEM images of 100 and 380 nm PSPs can be found in the Supporting Information of our previously published work. The size, dispersity, and suspension properties of the different-sized polystyrene particles are shown in Table S1 of the Supporting Information. All solutions were prepared using Nanopure water. UV–Vis spectra were recorded with a Shimadzu UV-2600i spectrophotometer (Duisburg, Germany). All measurements were carried out at ambient temperature in a UV-grade fused quartz cuvette with a 1 cm optical path length (Thorlabs).

2.2. Computational Methods

The computational model integrates single-particle electrodynamic calculations with Monte Carlo photon-transport simulations under experimentally matched boundary conditions. The framework is not intended as a purely parameter-free predictive theory, but as a physics-constrained forward modeling and parameter estimation approach that enables quantitative comparison with experimental measurements under realistic conditions.

2.2.1. Single-Particle Optical Response

Wavelength-resolved scattering cross sections were computed using Mie theory for homogeneous spherical particles. The particle diameter and the refractive indices of polystyrene and the surrounding aqueous medium were used as input parameters for calculating the optical properties of a single particle. For each wavelength, the model yields the total scattering cross section, the angular distribution of scattering intensity, and the polarization-dependent scattering amplitudes. The angular and polarization properties of the scattered light were determined from the Mie scattering matrix (Mueller matrix) elements along with the corresponding Poynting vector distributions. For nanoparticles of arbitrary shapes and compositions, numerical methods such as the discrete dipole approximation , will need to be implemented, and the optical responses averaged over different particle orientations in future work.

2.2.2. Simulation Geometry and Initial Conditions

Photon propagation was simulated within a cuvette (3 × 1 × 1 cm³), matching experimental dimensions. Incident photons were initialized at the center of the entrance face (Z = 0), propagating along the Z direction with linear polarization aligned along the X axis unless otherwise specified. The sample was modeled as a homogeneous particle suspension characterized by particle concentration and wavelength-dependent optical cross sections derived from Mie calculations.

2.2.3. Monte Carlo Photon Transport

Photon transport was modeled using a stochastic Monte Carlo algorithm. A detailed description is provided in the Supporting Information, with Figure S1 illustrating the simulation flowchart and Section S1.2 presenting the pseudocode. Each photon trajectory was propagated in discrete steps of 0.01 cm, a value chosen to be sufficiently small relative to the mean free path between scattering events across the full concentration range examined, ensuring convergence of detected intensity while maintaining computational efficiency. At each step, interaction probabilities were determined from the step size, particle concentration, and corresponding scattering cross sections.

For each photon trajectory step, a random number is used to determine whether the photon is scattered or transmitted based on the corresponding probabilities. In the simulations, these scattering and transmission probabilities are calculated using the Beer–Lambert law, given the particle concentration, the single-particle scattering cross section, and the step size which is 0.01 cm. The photon outcome at each step is then sampled by comparing a uniformly distributed random number to these probabilities. For example, if the transmission probability is 0.99 and the scattering probability is 0.01 for a given step, a photon will be scattered when a generated random number is less than 0.01; otherwise, it will be treated as transmitted. If the photon was transmitted without interaction, it continued propagating to the next step without modification. If a scattering event occurs, a new propagation direction and polarization state are sampled from the Mie-derived angular and polarization distributions. In the simulations, 10,000 candidate scattering directions are generated to achieve an approximately uniform distribution over the sphere by minimizing the repulsive energy of an equivalent system of point charges. These directions are precomputed along with their corresponding electric and magnetic fields and Poynting vector magnitudes (energy flux). A propagation direction is then selected by sampling from this distribution using a random number weighted by the energy flux. The corresponding electric and magnetic field orientations are subsequently assigned based on the Mie solution for the selected direction. This sequence was repeated iteratively until the photon reached the cuvette boundary. To ensure statistical robustness, simulations were repeated using different random-number seeds. For the reported results (≥10⁹ photon trajectories per condition), variations between independent runs were negligible, with deviations below 0.1% for the modeled UV–Vis spectra, confirming convergence and seed-independence of the final observables.

2.2.4. Boundary Conditions and Detection

When an incident or scattered photon reaches a cuvette boundary, refraction at the cuvette wall–environment interface is accounted for using Snell’s law and Fresnel coefficients, and the transmitted propagation direction and polarization state are updated accordingly. The cuvette wall thickness is taken as 0.125 cm, and its refractive index is assumed to be 1.5. When a photon exits the cuvette, its exit position, refracted propagation direction, and polarization state are recorded. Whether the photon contributes to the detected signal is then determined by geometric acceptance criteria corresponding to the experimental detector placement and collection angle.

For UV–Vis extinction simulations, only photons exiting within the defined forward detection solid angle, optimized using a supervised optimization model, were counted. Once determined, this angle would be fixed for all the simulations. In an idealized geometric description, the collection angle would be determined solely by the detector aperture and its distance from the cuvette. However, in the present experimental configuration, the effective detection angle is influenced by several additional factors that are not explicitly captured in the simplified geometric model. Specifically, the detected signal is affected by the finite detector aperture, lens alignment, angular sensitivity of the detector, beam divergence, etc. As a result, the “collection angle” in our model represents an effective angular acceptance parameter, rather than a strictly defined geometric angle. For polarization-resolved scattering simulations, detection windows were defined at specified sidewall locations for initial demonstration; these will be adapted to match experimental configurations in future work.

2.2.5. Machine Learning–Based Parameter Optimization Using Least-Squares Regression

The collection angle in the UV–Vis measurements, which cannot be determined precisely in experiment, was treated as an instrumental parameter to be learned from data. Here, “machine learning” is used in the broad sense of supervised regression. Specifically, parameter estimation was performed using a least-squares fitting approach rather than a neural network model.

Once optimized using this supervised regression framework, the collection angle was fixed across all simulations, ensuring consistent comparison with experimental data. In contrast, the wavelength-dependent refractive index of polystyrene was treated as an unknown physical parameter and was inferred through the same data-driven procedure, providing direct insight into the optical properties of the polystyrene nanoparticles.

A global loss function, defined as the mean squared deviation between simulated and experimental UV–Vis spectra across wavelengths, was minimized over data sets spanning multiple nanoparticle sizes and concentrations. The loss function is given by

$$L=\frac{1}{N}\sum _{i=1}^N{(y_i^{\exp }-y_i^{\mathrm{model}}(\theta ))}^2$$ 1

where θ represents the optimized parameters (e.g., collection angle or refractive index).

To improve robustness and mitigate overfitting, cross-validation was performed across data sets with different nanoparticle sizes, concentrations, and wavelength ranges, ensuring consistency of the optimized parameters across independent subsets.

It should be noted that the learned collection angle primarily serves as a calibration parameter for experimental geometry, while the inferred refractive index represents a physically meaningful material property. This approach enables quantitative agreement with experimental data without modifying the underlying photon transport framework, which continues to capture the essential physics of scattering and photon redistribution.

2.2.6. Statistical Sampling

For each wavelength and concentration condition, more than 5 × 10⁹ photon trajectories were simulated to ensure statistical stability in high optical density regimes. Each simulation at a single wavelength and concentration condition required over 20 h on a single AMD EPYC 7713 2.0 GHz CPU, making full spectral computation across multiple particle sizes, concentrations, and wavelengths tractable only through parallelization across independent wavelength and concentration conditions. Convergence was assessed by monitoring changes in detected intensity as a function of photon count. While the convergence depends on particle size, concentration, and wavelength, a variation below 0.1% in the collected UV–Vis spectra was adopted as the practical convergence criterion.

2.2.7. Spectral Construction

Simulated extinction and scattering spectra were constructed from the ratio of detected to incident photon counts, normalized to the corresponding reference condition without particles. Polarization-resolved intensities were computed by projecting the electric field vector of the exit photon onto defined polarization axes at the detection plane.

3. Results and Discussion

3.1. Validation of Measurement-Aware Modeling Against Experimental Spectra

The objective of this study is not solely to reproduce spectral trends, but to quantitatively assess whether deviations from Beer–Lambert extrapolation in the UV–Vis measurement can be explained by explicitly incorporating photon transport and experimental geometry without modifying intrinsic particle optical properties.

Figure a shows the schematic of the experimental setup. The incident light propagates along the Z-axis with polarization along the X-axis. The solution volume is 3 × 1 × 1 cm³, and collection angles along the forward Z-axis ranging from 1° to 5° were displayed. The optimized collection angle was determined to be 3°. Figure b–d compare experimentally measured UV–Vis spectra of 380 nm diameter polystyrene particles (PSP380) with predictions from (i) the measurement-aware model and (ii) Beer–Lambert extrapolation based on isolated-particle extinction cross sections and particle concentration. At low concentration (0.932 pM), both models agree with experimental data across the full wavelength range (300–800 nm). The root-mean-square error (RMSE) between the experimental data and the modeled results at a collection angle of 3° is below 0.0018, with a coefficient of determination (R ²) exceeding 0.9999. The effect of the collection angle on the spectrum is minimal at low concentrations, indicating that the dominant signal arises from the directly transmitted light of the incident beam.

2.

Comparison of experimental, modeled, and Beer–Lambert predictions of UV–Vis extinction spectra for PSP380. (a) Schematic of the measurement geometry. Incident light propagates along the Z-axis with polarization along X-axis. The cuvette dimensions are 3 × 1 × 1 cm³. The detector is positioned 36 cm from the cuvette, and the collection angle (θ) is treated as a model parameter. (b–d) Extinction spectra at particle concentrations of (b) 0.932 pM, (c) 11.0 pM, and (d) 21.9 pM. Experimental data (black circles) are compared with model predictions for different collection angles (θ = 1–5°, colored dashed lines) and the Beer–Lambert law (green solid line). Deviations from Beer–Lambert behavior increase at shorter wavelengths and higher concentrations.

At an intermediate concentration of 11.0 pM, deviations from Beer–Lambert predictions emerge at wavelengths below 400 nm. At 300 nm, the optical density calculated using the Beer–Lambert law exceeds 4.5, overestimating the measured extinction value of 2.8 by approximately 60%. The effect of instrumental parameters, such as the collection angle, on the detected signal becomes prominent: the UV–Vis extinction drops from the Beer–Lambert–predicted value of 4.5 to 3.1 at 1°, decreases to 2.7 at 3°, and further to 2.3 at 5°. The experimentally measured data and the results from the measurement-aware model with a collection angle of 3° maintain a root-mean-square error (RMSE) below 0.024 across the full spectral range.

At the highest concentration (21.9 pM), Beer–Lambert predictions reach optical densities approaching 9 at the wavelength of 300 nm, while the measured extinction remains below approximately 3. In this regime, Beer–Lambert extrapolation systematically overestimates extinction by more than 200%, whereas the measurement-aware model reproduces the experimental spectra with an RMSE below 0.025 across all wavelengths. Error analysis across the full wavelength range and concentration series is presented in Supporting Information Figure S2. The results indicate that the observed variations arise from concentration-dependent experimental factors as well as wavelength-dependent effects, potentially related to dispersion in the refractive index of polystyrene. The instrumental parameter, the collection angle, becomes increasingly critical: the UV–Vis signal drops from the Beer–Lambert law–predicted value of 9 to 4.2, even at a collection angle as small as 1°. The best agreement was also obtained at a collection angle of 3°, and this optimized value was retained for calculating the UV–Vis spectra of particles with other sizes.

These results demonstrate that the nonlinear concentration behavior in scattering-dominated systems can be quantitatively explained by photon redistribution and detection geometry, without invoking aggregation or changes to intrinsic optical constants. The quantitative agreement between the experimentally measured data and the measurement-aware model underscores the importance of considering instrumental parameters when interpreting experimental results using theoretical models.

3.2. Spatial Photon Redistribution and Nonlinearity

To identify the physical origin of the observed spectral distortion, spatial photon distributions within the cuvette were analyzed at the wavelength of 300 and 400 nm across increasing concentrations from 0.932 to 21.9 pM (Figure ). The incident light propagates along the Z-axis with polarization along the X-axis, and Figure a illustrates the collection geometry in the YZ plane.

3.

Simulation of light intensity and collection angle dependence for PSP380. (a) Schematic of the simulation setup defining the YZ plane within the cuvette. The incident light propagates along the Z-axis, with its polarization oriented parallel to the X-axis. (b–g) Simulated light intensity maps in the YZ plane for particle concentrations of (b, c) 0.932 pM, (d, e) 11.0 pM, and (f, g) 21.9 pM at wavelengths of 300 nm (top row) and 400 nm (bottom row). (h, i) Dependence of the measured UV extinction on the collection angle (0.25° to 25°) for varying concentrations from 0.932 to 21.9 pM at (h) the wavelength of 300 nm and (i) 400 nm.

Contour maps of photon distributions (Figure b–g) show that at low concentration (0.932 pM), detected photons arise predominantly from direct transmission, consistent with Beer–Lambert behavior shown in Figure b. As concentration increases, multiple scattering progressively redistributes photon trajectories, producing pronounced angular broadening of the photon flux. At optical densities above ∼5, more than 99% of detected photons originate from multiply scattered pathways rather than direct transmission.

This redistribution fundamentally alters the relationship between particle concentration and detected intensity. Because the detector samples a finite solid angle, the measured signal becomes strongly dependent on collection geometry and scattering anisotropy, rather than solely on the extinction cross section and concentration. Figure h,i quantify this effect, showing a pronounced dependence of detected intensity on collection angle, thereby highlighting the geometric contribution to the observed spectral distortion. As the collection angle increases, more rescattered photons are captured by the detector, and the detected extinction decreases accordingly. At the wavelength of 300 nm and concentration of 21.9 pM, the UV–Vis extinction predicted by the linear Beer–Lambert law is 9.1. However, the extinction calculated using the measurement-aware model decreases from 5.3 to 1.3 as the collection angle increases from 0.25° to 25°.

3.3. Mechanism Examination of Multiple Scattering Process

To validate the role of multiple scattering and its influence on the optical propagation of incident light in the cuvette, we calculated the average number of scattering events for all photon trajectories. A scattering event was counted each time an incident photon underwent a scattering interaction. While particles were initially treated as purely scattering particles in the simulation, absorption was subsequently introduced to investigate its effect on the underlying mechanism. In these simulations, the total extinction (defined as the sum of scattering and absorption) was kept unchanged relative to a PSP380 particle under the same conditions, while the absorption-to-extinction ratio (AER) was varied from a purely scattering system (AER 0.00) to a purely absorbing system (AER 1.00).

To further elucidate photon transport under multiple scattering in the cuvette, we analyze the number of scattering events per photon, the corresponding travel distance, and their impact on the detected spectrum, while varying the absorption fraction at fixed total extinction relative to a PSP380 particle under identical conditions. The evolution of the average number of scattering events is shown in Figure b. At high scattering fractions, a photon can undergo more than 25 scattering events before exiting the cuvette. Introducing absorption reduces the number of rescattering events, which approaches zero in the purely absorbing limit.

4.

(a) Schematic illustration of a photon undergoing multiple scattering events in a particle suspension before exiting the cuvette. (b) Average number of scattering events per photon and (c) average photon travel distance as functions of optical density (Beer Lambert UV extinction) for different AER levels, while keeping the total extinction cross section unchanged relative to a PSP380 particle under identical conditions. (d–f) Simulated UV–Vis spectra at particle concentrations of (d) 0.932 pM, (e) 11.0 pM, and (f) 21.9 pM for varying AER conditions, together with experimentally measured spectra (Black dots).

We also calculated the average photon path length within the cuvette. As shown in Figure c, increasing absorption reduces the photon path length. Multiple scattering leads to an effective photon travel distance that exceeds the physical path length of 1 cm by more than 30% under purely scattering conditions at an optical density of 9, reaching approximately 1.3 cm. This extension of the optical path is important for applications such as fluorescence detection, as it enhances the probability of excitation by increasing the interaction length between light and molecules.

Notably, an additional path length of 0.3 cm corresponds to a temporal delay of approximately 10 ps. While this delay is negligible for nanosecond-scale fluorescence lifetimes, it becomes relevant for ultrafast processes in the picosecond regime, where photon transport time can contribute to the measured temporal response and should be considered in high-resolution lifetime measurements.

Figure d–f shows the evolution of the modeled extinction spectrum as the absorption fraction is varied from 0% to 100% while keeping the total extinction cross section unchanged. At a concentration of 0.932 pM (Figure d), scattering is minimal, and the detected signal is dominated by directly transmitted light rather than rescattered photons. Consequently, varying the absorption fraction has a negligible effect on the measured extinction.

At a concentration of 11.0 pM (Figure e), the contribution from rescattered light becomes significant. As the absorption fraction increases from 0% to 100%, the detected extinction (within a 3° collection angle) increases from 2.7 to 4.5 at 300 nm wavelength. The simulated spectrum agrees well with experimental measurements in the purely scattering limit (AER 0.00, 0% absorption) and converges to the prediction of the Beer–Lambert law in the purely absorbing limit (AER 1.00, 100% absorption).

This effect becomes more pronounced at higher concentrations. At 21.9 pM (Figure f), the extinction increases from 3.0 to 9.0 at 300 nm wavelength as the AER increases from 0% to 100%. As in the 11.0 pM case, the model agrees with experimental measurements in the absence of absorption and approaches the Beer–Lambert prediction in the fully absorbing limit.

3.4. Polarization Redistribution and Depolarization Metrics

Polarization-resolved simulations further reveal geometry-dependent modifications of the scattered photons (Figure ). Please note that these results are presented as a model-based demonstration of the predicted polarization dependence within the current theoretical framework, rather than as direct experimental validation. The depolarization ratio, defined as ρ = I _Ez/I _Ex at the detection plane, increases from nearly zero at 0.932 pM to >0.22 at 21.9 pM when the detection window is positioned at the center of the cuvette sidewall. Notably, the depolarization, ρ, increases substantially when the detection window is shifted away from the sidewall center, highlighting the strong spatial dependence of polarization mixing within the cuvette geometry.

5.

Simulation of scattering intensity and depolarization for PSP380 at a wavelength of 300 nm. (a) Schematic of the experimental simulation setup. The incident light propagates along the Z-axis, with its polarization oriented parallel to the X-axis. For the initial exploration, the signal collection occurs at the cuvette sidewall in the XZ plane. Windows 1–5 represent discrete collection positions in the same plane and translated along the Z (k) axis. (b–j) Simulated scattering intensities in the XZ plane at the cuvette sidewall for increasing concentrations (columns). Rows represent intensity components parallel to the X-axis (I _Ex), Z-axis (I _Ez), and the resulting depolarization maps (I _Ez/I _Ex). (k–m) Extracted scattering intensities and depolarization values for windows 1–5 (as defined in panel a) across the full concentration range.

Figure a shows the schematic of the simulation setups. The incident beam passes through the cuvette along the Z-axis with its polarization parallel to the X-axis, and signal collection occurs at the cuvette sidewall in the XZ plane. Windows 1–5 represent discrete collection positions translated along the Z (k) axis in the same plane with a collection area of 0.05 × 0.05 cm². The simulation configurations are intended for initial exploration and will be adjusted to match experimental conditions in future studies.

Panels (b–j) show the calculated scattering intensity distributions in the XZ plane at the cuvette sidewall for three representative concentrations (0.932, 11.0, and 21.9 pM). The top row corresponds to the X-polarized component (I _Ex), and the middle row shows the Z-polarized component (I _Ez), and the bottom row displays derived depolarization ratio (I _Ez/I _Ex).

At the lowest concentration (0.932 pM), multiple scattering events occur; however, they are not dominant, as evidenced by the low depolarization ratio, I _Ez/I _Ex (panel d) with the lowest value of 0.01 at the center and values up to 0.52 near the edges. As the concentration increases approximately 12-fold to 11.0 pM, and further to 21.9 pM, a pronounced enhancement in the total scattering intensity is observed for both polarization components. The Z-polarized component increases disproportionately relative to the X-polarized component as multiple scattering progressively randomizes photon propagation directions, transferring intensity into the cross-polarized channel. This transition from single- to multiple-scattering behavior is clearly reflected in the depolarization ratio maps (panels d, g, and j), where both the magnitude and spatial extent of high-depolarization regions increase substantially with concentration. The most pronounced change occurs between 0.932 pM and 11.0 pM, where the minimum depolarization increases from 0.01 to 0.10 and the maximum from 0.52 to 1.74. At 21.9 pM, the minimum depolarization further increases to 0.22, while the maximum increases only slightly to 1.80. We further observe that the dominant scattering direction shifts progressively toward the backward direction with increasing concentration, consistent with the growing contribution of multiple scattering to photon transport.

Panels (k–m) quantify the concentration- and position-dependent scattering behavior by extracting intensity and depolarization values from five predefined detection windows along the Z-axis (Figure a). For X-polarized photons (Figure k), the intensity at Window 1 increases monotonically with concentration, consistent with the expected rise in scattering probability as particle concentration increases. A similar trend is observed at Window 2, which records a higher photon count due to stronger forward-directed scattering.

At Window 3, however, the intensity exhibits a nonmonotonic dependence on concentration, reaching a maximum near ∼15 pM before decreasing at higher concentrations. As the detection window moves further along the forward (Z) direction, this intensity maximum shifts toward lower concentrations and its magnitude progressively decreases. This behavior indicates the onset of significant multiple scattering at elevated concentrations. While scattering increases with concentration at low particle densities, further increases in concentration give rise to multiple scattering that redistributes photons angularly, resulting in a reduction of the forward-directed intensity detected in a cuvette geometry.

For Z-polarized photons (Figure l), the intensity generally scales with concentration across all windows, reflecting the increasing contribution of polarization-mixing processes. A pronounced spatial dependence is observed: the intensity increases from Windows 1 to 3, consistent with progressive polarization conversion along the photon propagation path, and then decreases at Windows 4 and 5, likely due to angular redistribution of the scattered photons.

The depolarization ratio (Figure m) exhibits a concentration-dependent spatial behavior. At low concentration, the depolarization decreases nearly monotonically from Windows 1 to 5. For example, at 0.932 pM, the depolarization decreases from approximately 0.1 at Window 1 to nearly zero (0.01) at Window 5, indicating that polarization is largely preserved in the forward propagation direction under weak scattering conditions, whereas backward scattering contributes more significantly to depolarization.

As the concentration increases, spatial dependence becomes nonmonotonic: at the highest concentration of 21.9 pM, depolarization decreases from 0.45 in Window 1 to 0.23 in Window 3, then increases in Windows 4 and 5 to 0.28 and 0.35, respectively. This behavior reflects the growing influence of multiple scattering, which enhances polarization mixing at larger propagation distances and leads to a more complex spatial redistribution of the depolarized component.

The simulations reveal that depolarization is strongly spatially dependent and highly sensitive to the detection geometry, underscoring the critical importance of window placement in experimental measurements. These results demonstrate that, in the multiple-scattering regime, polarization observables encode the cumulative photon transport history rather than solely single-particle scattering properties. Furthermore, these findings support the use of spatially and polarization-resolved detection as a sensitive probe of concentration-dependent scattering dynamics, providing enhanced information about particles in solutions.

3.5. Cross-Validation across Particle Sizes

To evaluate generality, the model was applied to PSP100, PSP380, and PSP1060 across 24–30 concentration conditions (Figure ). Figure panels (a–c), (d–f), and (g–i) correspond to PSP100, PSP380, and PSP1060, respectively. Across all particle sizes and wavelengths (350–800 nm; Figure a,d,g), the measurement-aware model reproduces the experimental extinction spectra with root-mean-square errors below 0.017, 0.024, and 0.057 for the three particle sizes at a collection angle of 3°. For clarity, only every other concentration value is shown. The standard deviation of the measured spectra is about 0.002, which is much lower than the root-mean-square errors, indicating that the contribution of measurement noise to the overall error is negligible. A detailed standard deviation analysis for experimental spectra collected at different particle sizes and concentrations is provided in Supporting Information Section S4.

6.

Experimental, modeled, and Beer–Lambert law–predicted UV–Vis extinction for (a–c) PSP100, (d–f) PSP380, and (g–i) PSP1060. (a, d, g) Experimentally measured and theoretically modeled extinction spectra as a function of wavelength for varying concentrations. (b, e, h) Comparison of measured and theoretically modeled data against the linear Beer–Lambert law (blue line) at the wavelength of 400 nm. (c, f, i) Comparison across the full spectral range (350–800 nm), highlighting deviations of the measured and modeled data from linear behavior predicted using Beer–Lambert law. Dashed vertical lines in (a, d, g) indicate the wavelength (400 nm) used to extract the data shown in (b, e, h). The shaded green regions in (c, f, i) indicate the threshold beyond which deviations become significant.

At the wavelength of 400 nm (Figure b,e,h), the extinction follows Beer–Lambert linearity for small particles (PSP100) at low optical densities but progressively deviates as particle size and concentration increase, consistent with the growing influence of multiple scattering and interparticle interactions. Figure c,f,i compile all data points across the wavelength range of 350–800 nm. The full spectral analysis confirms that Beer–Lambert linearity is maintained for smaller particles and low optical densities but increasingly breaks down for larger particles, particularly at wavelengths where scattering is the strongest. The shaded green regions indicate the thresholds beyond which deviations from linearity become significant. For PSP100, linearity is maintained up to an optical density of 3, whereas for PSP1060, deviations gradually increase beyond an optical density of 1.2, indicating strong size dependence. Nevertheless, the model successfully reproduces these nonlinear trends across the full spectral range, supporting its applicability to concentrated and strongly scattering suspensions.

For PSP1060, optimal agreement was achieved only when the effective particle diameter was adjusted to 970 nm, corresponding to an approximately 8.5% deviation from the vendor specification. We emphasize that this adjustment should be interpreted as an effective optical diameter, rather than a change in the physical particle size.

This discrepancy likely reflects the sensitivity of the optical response in the micron-size regime to interfacial and boundary effects, including surface hydration, polymer–water interfacial layers, and associated modifications of the local dielectric environment. Since the complex effective dielectric constant (ε ≈ (n + ik)² for nonmagnetic media, where n and k are the real and imaginary components of the refractive index) directly determines the boundary conditions − for electromagnetic scattering, even small deviations at the particle–medium interface can lead to measurable shifts in resonant scattering features.

For smaller nanoparticles, whose optical response is dominated by dipolar excitation modes, such surface-related modifications contribute minimally to the overall scattering cross section. In contrast, for larger nanoparticles with sizes on the order of one micron, whispering gallery modes are highly sensitive to changes in the surface dielectric environment; even modest interfacial modifications can therefore produce measurable discrepancies between experimentally observed and theoretically predicted spectra. Once the particle size is fixed, the spectra at all concentrations were reproduced without further parameter adjustment, indicating that the discrepancies originate from effective optical parameters rather than concentration-dependent model artifacts.

3.6. Parameter Identifiability and Uncertainty Analysis

Accurate modeling of scattering-dominated systems requires that intrinsic optical parameters be identifiable independently of transport-induced distortion. To assess parameter identifiability, we evaluated sensitivity of the modeled spectra to variations in particle diameter and refractive index.

Perturbation analysis shows that a ±1% variation in particle diameter produces spectral deviations of up to ±2% near scattering maxima, while an equivalent variation in refractive index induces wavelength-dependent deviations as high as 13%. These sensitivities exceed the residual modeling error, indicating that both parameters can be reliably identified within the experimental noise level. A supervised regression model based on least-squares fitting algorithm was trained across multiple concentration data sets to predict refractive indices from measured spectra. Confidence intervals were computed from the model residuals to quantify uncertainty in the extracted values. As shown in Figure , extracted refractive index values from different particle sizes agree within ±3%, suggesting slight size-dependent variations in the effective refractive index of polystyrene, which may reflect intrinsic material heterogeneity or subtle surface effects. We treated polystyrene particles as purely scattering particles in the visible range. The agreement between experimentally measured and theoretically modeled spectra across different concentrations and wavelengths supports this assumption. In absorption-dominated systems, extinction is linearly proportional to concentration and follows the Beer–Lambert law. In contrast, scattering introduces deviations from this linear behavior due to multiple scattering effects. The observed agreement across concentrations therefore supports the validity of treating polystyrene as a purely scattering material. Please note that the extracted parameters should not be interpreted as fundamental material constants, but rather as effective quantities that provide insight into particle–environment interactions. Importantly, parameter estimates remained stable across concentration regimes where Beer–Lambert extrapolation fails, demonstrating that inclusion of photon transport mitigates geometry-induced inference bias.

7.

Effective refractive index of polystyrene as a function of wavelength from 300 to 800 nm. Dashed lines represent calculated values based on three particle diameters: 100 nm (black), 380 nm (red), and 1060 nm (green). The solid blue line represents the averaged index of refraction, n, over three sized nanoparticles.

3.7. Implications for Measurement Modeling

Collectively, these results demonstrate that Beer–Lambert extrapolation introduces systematic inference error in scattering-dominated systems, even when experimental spectra exhibit excellent signal-to-noise ratio and fall well within the instrument’s linear dynamic range. The apparent reliability of high-quality spectral data therefore does not guarantee physical accuracy when multiple scattering and transport effects are neglected.

These limitations are not confined to polystyrene particle suspensions but are expected to arise broadly in any optically dense scattering medium. Practically relevant systems include protein aggregates and amyloid fibrils, where scattering contributions to UV–Vis spectra are frequently attributed to changes in secondary structure or aggregation state; − liposomes and lipid nanoparticles, where size and encapsulation efficiency are routinely inferred from extinction spectra; , metal nanoparticle colloids, where plasmonic peak positions and line widths are sensitive to multiple scattering and interparticle coupling at finite concentrations; ,,− and biological tissue phantoms and turbid suspensions used in clinical and diagnostic optical measurements. ,,

In all of these systems, the same geometry-dependent photon redistribution effects demonstrated here will introduce concentration-, wavelength-, and size-dependent deviations from isolated-particle predictions that cannot be corrected by simple empirical factors. However, it is important to note that the photon transport framework employed in this work treats particles as independent scatterers. While absorption can be readily included in the model, variations in particle shape and composition, such as those encountered in the systems listed above, are not explicitly considered, and near-field interparticle coupling effects (e.g., plasmonic interactions in closely spaced metal nanoparticles) are not captured. In such cases, coupling modifies the single-particle optical response itself, which must be accounted for prior to applying the transport model, for example through numerical electromagnetic approaches such as the discrete dipole approximation. , Additionally, for nonspherical particles, orientation-dependent optical responses would need to be averaged at the single-particle or -cluster level. We note that it is somewhat surprising that the simulations achieve quantitative agreement with experimental measurements across different particle sizes, concentrations, and wavelengths, given that the current model does not explicitly account for size distributions in the system. Furthermore, the different refractive indices derived for particles of different sizes suggest the presence of material inhomogeneity, providing new physical insight enabled by the developed model. Thus, while the present approach captures multiple scattering and photon redistribution in dense systems, additional modeling is required for particles with complex shapes, varied compositions, or broad size distributions, as well as in regimes where collective electromagnetic interactions between particles become significant.

Importantly, the findings also show that intrinsic optical parameters can still be extracted in a consistent and physically meaningful manner when transport effects are explicitly modeled. Incorporating photon transport into the forward model restores parameter identifiability and prevents systematic bias, a principle that applies equally to the extraction of plasmonic resonance parameters, protein aggregate size distributions, and lipid nanoparticle refractive indices from ensemble UV–Vis measurements.

Together, these conclusions establish a quantitative foundation for integrating electrodynamic theory with measurement-aware photon transport modeling in optical spectrometry, ensuring that inferred material properties remain physically grounded under realistic experimental conditions across a broad range of materials and measurement contexts.

4. Conclusion

In this work, we established a foundational, measurement-aware computational framework for quantitative interpretation of UV–Vis spectra in scattering-dominated particle systems. By explicitly integrating electrodynamic theory with Monte Carlo photon-transport simulations and supervised machine-learning optimization of instrumental parameters, the framework treats photon transport, multiple scattering, and experimental geometry as intrinsic components of the measurement process rather than as artifacts or sources of experimental error. Using spherical polystyrene particles as a model system, we demonstrated that experimentally observed spectral distortion and nonlinearity, often attributed to changes in particle structure or aggregation, can be quantitatively reproduced without invoking modifications to intrinsic material properties. Across a wide range of particle sizes, concentrations, and wavelengths, the framework successfully reproduces experimentally measured UV–Vis extinction spectra, including regimes exhibiting strong deviations from Beer–Lambert behavior. Spatial and polarization-resolved simulations reveal that cascading photon–matter interactions fundamentally redistribute photon intensity and polarization, providing a physical origin for nonlinear extinction and apparent saturation effects even within nominal instrumental linear regimes. Importantly, by leveraging these interactions as information-bearing constraints, the model enables direct extraction of effective dielectric constants with consistency across independent data sets.

The sensitivity of the framework to effective optical parameters, as demonstrated for larger polystyrene particles, highlights its capacity to expose subtle deviations between nominal and solution-phase optical properties that would remain obscured under conventional isolated-particle analysis. Rather than representing a limitation, this sensitivity underscores the value of treating photon transport and multiple scattering as diagnostic tools for material characterization. As the first installment of a broader computational series, this work establishes the scattering-only foundation of a unified strategy for modeling optical spectroscopy in complex materials. The principles demonstrated here provide a necessary baseline for subsequent extensions to systems exhibiting coupled absorption and scattering, fluorescence and reabsorption, and ultimately fully correlated absorption–scattering–emission dynamics. By progressively incorporating increasing photophysical complexity within a consistent measurement-aware framework, this approach offers a general pathway toward physically grounded, quantitative interpretation of optical spectra across a wide range of materials and environments.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c03515.

• Monte Carlo Simulations (S1); Vendor-Provided Specifications and Suspension Properties of the Polystyrene Particle Samples (Polysciences, Inc.) (S2); Wavelength- and Concentration-Dependent Error Analysis Between the Experimentally Measured and Theoretically Modeled Extinction Spectra of PSP380 (S3); and Standard Deviation Analysis for Experimental UV–Vis Spectra (S4) (PDF)

The authors declare no competing financial interest.